TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 9, September 1998, Pages 3571–3585 S 0002-9947(98)02202-8

INDUCTION THEOREMS ON THE STABLE RATIONALITY OF THE CENTER OF THE RING OF GENERIC MATRICES

ESTHER BENEISH

Abstract. Following Procesi and Formanek, the center of the division ring of n × n generic matrices over the complex numbers C is stably equivalent to the fixed field under the action of S_{n}, of the function field of the group algebra of a ZS_{n}-lattice, denoted by G_{n}. We study the question of the stable rationality of the center C_{n} over the complex numbers when n is a prime, in this module theoretic setting. Let N be the normalizer of an n-sylow subgroup of S_{n}. Let M be a ZS_{n}-lattice. We show that under certain conditions on M, inductionrestriction from N to S_{n} does not affect the stable type of the corresponding field. In particular, C(G_{n}) and C(ZG ⊗_{ZN} G_{n}) are stably isomorphic and the isomorphism preserves the S_{n}-action. We further reduce the problem to the study of the localization of G_{n} at the prime n; all other primes behave well. We also present new simple proofs for the stable rationality of C_{n} over C, in the cases n = 5 and n = 7.

Introduction

The question of rationality of the center of the ring of generic matrices has been studied extensively over some decades, in particular for its connections to other problems. Geometrically, the center C_{n} of 2 n × n generic matrices over an algebraically closed field is the function field of an algebraic variety under a reductive group, namely (M_{n} × M_{n})/ PGL_{n}. The classification of quotients under connected reductive groups is one of the main open problems in invariant theory. In the Brauer group setting the stable rationality of this center implies the Merkurjev-Suslin result, which says that the Brauer group of a field k is generated by cyclics, provided k has enough roots of unity. Le Bruyn presents an informative survey of the problem and its applications in [LL].

In 1883 Sylvester showed that for n = 2, C_{2} was rational over C, the complex numbers, [S]. In 1979 and 1980 Formanek showed that C_{n} was rational over C for the case n = 3, [F1], and n = 4, [F2]. In 1984, Saltman proved that for all primes n, C_{n} is retract rational over C, [SD]. In 1991 Bessenrodt and Le Bruyn showed stable rationality for the cases n = 5 and 7 [BL]. Their proof, however, requires the use of a computer. In this paper we present simple proofs for the case n = 5 and 7, Theorems 3.3 and 3.4.

We study the problem in its module theoretic setting. If G is a finite group, and M is a ZG-lattice, then C[M] denotes the group algebra of the abelian group M, and C(M) its quotient field. Define the following ZS_{n}-lattices. U_{n} = ∑_{i=1,...,n} Zu_{i}, where gu_{i} = u_{g(i)} for all g in S_{n}. A, usually denoted by A_{n−1}, is the kernel of the

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors September 22, 1996.

1991 Mathematics Subject Classification. Primary 13A50, 20C10.

augmentation map ε: U_{n} → Z. V_{n} = ∑_{i,j,i≠j} Zy_{ij} where gy_{ij} = y_{g(i)g(j)} for all g in S_{n}. G_{n} is defined by the sequence 0 → G_{n} → V_{n} → A → 0, where the map V_{n} → A is given by y_{ij} → u_{i} − u_{j}. Procesi-Formanek show that C_{n} ≅ C(G_{n} ⊕ U_{n} ⊕ U_{n})^{Sn}, [F1, Theorem 3]. This implies that C_{n} is stably isomorphic to C(G_{n})^{Sn} [L, Proposition 1.4]. Let n be an odd prime. Let H be an n-sylow subgroup of G, thus H is cyclic of order n. The normalizer of H in S_{n}, which we denote by N, is the semidirect product of H by a cyclic group C of order n − 1.

The main result of this paper is:

Theorem 2.6. The fields C(G_{n}) and C(ZG ⊗_{ZN} G_{n}) are stably isomorphic, and the isomorphism respects the S_{n}-actions. The center C_{n} is stably isomorphic to their fields of invariants.

This theorem is a direct consequence of Proposition 2.4, which says that under certain conditions, induction—restriction from N to G does not affect the stable type of the field. This is a crucial reduction step for the problem. It implies that the stable rationality of C_{n} depends on the structure of G_{n} as a ZN-lattice. Set n = p, and let Z_{p} denote the localization of Z at the prime p.

The second result is:

Theorem 2.7. Let D and R be invertible ZN-lattices. Suppose that R is either 0, or stably permutation and Z_{q}C-projective for all primes q dividing p − 1. If (G_{n} ⊕ R)_{p} ≅ D_{p}, then the fields C(G_{n}) and C(ZG ⊗_{ZN} D) are stably isomorphic as G-fields.

This theorem says that the stable rationality of C_{n} in fact depends on the structure of (G_{n})_{p} as a Z_{p}N-lattice.

I would like to thank Professors John Moody, Robert Guralnick and David Saltman, for their support and advice.

1.

Let G be a finite group, a ZG-lattice is finitely generated Z-free ZG-module. A permutation ZG-module is a ZG-lattice with a Z-basis permutated by G. An invertible ZG-module is a direct summand of a permutation module. A ZG-module M is stably permutation if there exists permutation ZG-modules P and P′, such that M ⊕ P ≅ P′.

Definition. A ZG-lattice M is G-faithful if the map G → Aut_{Z}(M) is injective.

Notation.

(1) For a ZG-lattice M we denote by M^{∗} = Hom_{Z}(M, Z), its dual.

(2) For any prime q in Z, Z_{q} denotes the localization of Z at the ideal (q), and M_{q} = Z_{q} ⊗ M.

(3) In the remainder of this paper we will denote S_{n} by G, and n = p will be an odd prime, unless otherwise specified.

The following observations were made in [BL].

The map: V_{n} → A ⊗ U_{n}

y_{ij} → (u_{i} − u_{j}) ⊗ u_{i}

is a ZG-isomorphism.

Now tensoring the exact sequence

0 → A → U_{n} → Z → 0

by A, we get

0 → A ⊗ A → U_{n} ⊗ A → A → 0

The map U_{n} ⊗ A → A sends y_{ij} to u_{i} − u_{j}, so G_{n} ≅ A ⊗ A.

Notation. We let B = A^{∗} ⊗ A.

This lattice is used throughout the paper, in particular if let Q be the field of rational numbers, then QB ≅ QG_{n}.

The following proposition was proved by Bessenrodt and Le Bruyn, but not published. We present here a proof.

Proposition 1.1. We have the following isomorphisms of ZG-lattices B ⊕ U_{n} ≅ V_{n} ⊕ Z. Hence B is stably permutation.

Proof. The sequence 0 → A → U_{n} → Z → 0 splits when localized at all primes q different from n, dividing the order of G, with splitting map given by 1 → (∑_{i−1}^{n} u_{i})/n. So we have

Z_{q} ⊕ A_{q} ≅ (U_{n})_{q} and taking duals Z_{q} ⊕ A_{q}^{∗} ≅ (U_{n})_{q}.

Now tensoring by A_{q} we get

B_{q} ⊕ A_{q} ≅ (V_{n})_{q}.

So B_{q} is Z_{q}G-invertible. In particular B_{q} is Z_{q}P -invertible for all q-sylow subgroups P of G. To show that B is invertible it suffices by [CT, Lemma 9, section 1] to show that B_{p} is Z_{p}H-invertible for a p = n-sylow subgroup of G. Now consider the diagram

\xymatrix{ &  & 0 & 0 \\ 0 \ar[r] & B \ar[r] & V_{n} \ar[r] \ar[u] & A^{∗} \ar[r] \ar[u] & 0 \\ 0 \ar[r] & B \ar[r] \ar[u] & X \ar[r] \ar[u] & U_{n} \ar[r] \ar[u] & 0 \\  &  & Z \ar[r] \ar[u] & Z \ar[u] \\  &  & 0 \ar[u] & 0 \ar[u]}

(U_{n})_{p} ≅ Z_{p}S_{n}/S_{n−1} and (V_{n})_{p} ≅ Z_{p}S_{n}/S_{n−2} are Z_{p}G-projective, since they are induced from groups of order prime to p, hence X_{p} ≅ (B ⊕U_{n})_{p} ≅ (V_{n} ⊕Z)_{p}. So B_{p} is Z_{p}G-invertible, and hence Z_{p}H-invertible. So B is ZG-invertible. This implies by e.g. [B, Lemma 1.2]

B ⊕ U_{n} ≅ V_{n} ⊕ Z.

So B is stably permutation.

Lemma 1.2. Let X = Z/pZ be the trivial ZG-module with p elements. Then there exists a ZG-exact sequence

0 → A → A^{∗} → X → 0.

Proof. U_{n}^{∗} is generated by u_{i}^{∗}, where u_{i}^{∗}(u_{j}) = δ_{ij}, and A^{∗} is generated by res(u_{i}^{∗})I = 1, …, n − 1, where res is restriction from U_{n} to A. The map A^{∗} → X is given by res(u_{i}^{∗}) → 1.

Remark. Tensoring the sequence of Lemma 1.2 by A we get

0 → A ⊗ A → A^{∗} ⊗ A → X ⊗ A → 0.

Identifying A ⊗ A with G_{n}, and setting T = X ⊗ A ≅ A/pA we obtain

0 → G_{n} → B → T → 0.

Let U = U_{n}/pU_{n}. It is easy checked that the map U → X given by u_{i} → 1 has kernel T. Thus we have a ZG-exact sequence

0 → T → U → X → 0.

Let T′ = A^{∗}/pA^{∗}, then it is immediate that the following sequence is ZG-exact

0 → X → U → T′ → 0.

These are important sequences which will be used in the following sections.

2.

Throughout the rest of this paper we adopt the following notation unless otherwise specified. We will also keep the notation used in section 1.

• G = S_{n}, n = p is prime; n and p will be used interchangeably without mention.

• H = p-sylow subgroup of G, so H is cyclic of order p.

• N = N_{G}(H) will be the normalizer of H in G, so N = H ⋊ C, where C is cyclic of order p − 1, and if we let h and c generate H and C respectively, then chc^{−1} = h^{a} where a is a primitive (p − 1)st root of 1 mod p.

• F = Z/pZ field of p elements.

• For any finite group G, and for any ZG-lattice M, we will denote by ϕ(M) the flasque class of M, see [CT, section 1] for definitions.

• Z^{−} will denote the sign representation of G.

• X will denote the trivial ZG-module of p elements.

Definitions. (1) A ZG-lattice M is quasi-permutation [EM], if there exist an exact sequence

0 → M → R → Q → 0

with R and Q permutation.

(2) We will say that 2 fields are stably isomorphic as G-fields, if the fields are stably isomorphic, and the isomorphism respects their G-actions.

Various versions of the following lemma have been proved in the literature, we need a slightly different version here.

Lemma 2.1. Let G be a finite group.

(1) Suppose there exists an exact sequence of ZG-lattices

0 → M → N → E → 0

with E invertible and M, G-faithful, then C(E ⊕ M) ≅ C(N) as G-fields.

(2) Let G be a finite group, and let M and M′ be G-faithful ZG-lattices, with M quasi-permutation. Then the fields C(M ⊕ M′) and C(M′) are stably isomorphic as G-fields.

(3) Let M and M′ be G-faithful ZG-lattices such that ϕ(M) = ϕ(M′), then C(M) and C(M′) are stably isomorphic as G-fields.

(4) If G = S_{n} and M is a G-faithful quasi-permutation ZG-lattice, then C(M)^{G} is stably rational over C.

Proof. (1) The proof is basically that of [L, Proposition 1.5]. The injection M → N induces an injection C(M) ⊂ C(N). Now consider the sequence

0 → C(M)^{∗} → C(M)^{∗}N → E → 0.

Since G acts faithfully on C(M)^{∗}, C(M)^{∗} is H^{1}-trivial by Hilbert theorem 90. Since E is invertible the sequence splits by [CT, Lemma 1, section 1]. Thus C(M)^{∗}N ≅ E ⊕ C(M)^{∗}. This isomorphism yields the required isomorphism of G-fields.

(2) Since M is quasi-permutation there exist an exact sequence

0 → M → R → Q → 0

with Q and R permutation. By (1) C(M ⊕ Q) ≅ C(R). So C(M ⊕ M′ ⊕ Q) ≅ C(M′ ⊕ R). Apply [L, Proposition 1.3] to the field l = C(M′) and the l-vector space W = ∑ lr_{i}, where {r_{i}} is a Z-basis for R which is permuted by G. Then there exists elements {y_{i}} in W^{G} which form an L-basis for W. Now W ⊂ l(R), and hence

l(R) = l(y_{1}, …, y_{k}) = C(M′)(y_{1}, …, y_{k}).

By the same argument there exist indeterminates z_{1}, …, z_{t}, such that

C(M ⊕ M′ ⊕ Q) = C(M ⊕ M′)(z_{1}, …, z_{t}).

So C(M′)(y_{1}, …, y_{k}) ≅ C(M ⊕ M′)(z_{1}, …, z_{t}).

(3) Since ϕ(M) = ϕ(M′), there exist 2 exact sequences

0 → M → E → R → 0 and 0 → M′ → E → P → 0

with P and R permutation by [CT, Lemma 8, section 1]. By (1) C(E) ≅ C(M ⊕ R) ≅ C(M′ ⊕ P) as G-fields, and by (2) C(M) ad C(M′) are stably isomorphic as G-fields.

(4) Since M is quasi-permutation there is an exact sequence

0 → M → R → Q → 0

with Q and R permutation. By (1) C(M ⊕ Q) ≅ C(R). By (2) C(M ⊕ Q) and C(M) are stably isomorphic as G-fields. Since ϕ(U_{n}) = ϕ(R) = 0, C(U_{n}) and C(R) stably isomorphic as G-fields by (3). Since C(U_{n})^{G} is rational over C, the result follows.

Proposition 2.2. (1) Let G be a finite group and suppose there exist exact sequences

0 → E → P → L → 0 and 0 → E′ → R → J ⊕ L → 0

with P and R, ZG-permutation, E and E′ G-faithful, L and J, finite and J = Q/mQ for some ZG-permutation module Q, and some positive integer m. Then the fields C(E′) and C(E) are stably isomorphic as G-fields.

(2) Let G = S_{n} and let M be a G-faithful ZG-lattice of trivial cohomology, then C(M)^{G} is stably rational over C. Furthermore if M′ is any G-faithful ZG-lattice, then the fields C(M ⊕ M′) and C(M′) are stably isomorphic as G-fields.

Proof. (1) We form the commutative diagram

\xymatrix{ &  & 0 & 0 \\ 0 \ar[r] & E′ \ar[r] & R \ar[u] \ar[r] & J ⊕ L \ar[u] \ar[r] & 0 \\ 0 \ar[r] & E′ \ar[u] \ar[r] & M_{3} \ar[u] \ar[r] & Q ⊕ P \ar[u] \ar[r] & 0 \\  &  & Q ⊕ E \ar[u] \ar[r] & Q ⊕ E \ar[u] \\  &  & 0 \ar[u] & 0 \ar[u]}

Since R, Q and P are permutation modules, the fields C(E) and C(E′) are stably isomorphic as G-fields, by Lemma 2.1.

(2) By [BK, Theorem 8.10] M is ZG-projective and by [EM, Theorem 3.3] M is stably permutation. The lemma follows by Lemma 2.1.

Notation. Let I_{G/N} denote the kernel of the augmentation map Z(G/N) → Z, and let I = I_{G/N}/pI_{G/N}.

Proposition 2.3. The Z_{p}G-lattice (I_{G/N})_{p} and the ZG-lattice I are cohomologically trivial. Let I_{G/N}^{−} = I_{G/N} ⊗ Z^{−}, I^{−} = I ⊗ Z^{−}, then (I_{G/N}^{−}), I^{−} are also cohomologically trivial.

Proof. The sequence 0 → (I_{G/N})_{p} → Z_{p}(G/N) → Z_{p} → 0 splits with splitting map Z_{p} → Z_{p}(G/N) given by 1 → (∑_{g∈G/N} g)/[G : N]. So Z_{p}(G/N) ≅ Z_{p} ⊕ (I_{G/N})_{p}. By Mackey’s subgroup theorem [CR1, Theorem 10.13] Res_{H}^{G} ZG/N ≅ Z ⊕ ZH^{k} for some positive integer k. So as a Ẑ_{p}H-module (Î_{G/N})_{p} is free by the Krull-Schmidt-Azumaya and by [CR1, Theorem 30.17] (I_{G/N})_{p} is a free Z_{p}H-module. Now for any subgroup K of G, H^{i}(K, (I_{G/N})_{p}) injects into H^{i}(H, (I_{G/N})_{p}) = 0. This proves that (I_{G/N})_{p} is cohomologically trivial. To prove that I is cohomologically trivial it suffices to take the cohomology of the sequence

0 → (I_{G/N})_{p} → (I_{G/N})_{p} → I → 0.

The second statement follows since I_{G/N}^{−}, I^{−} are isomorphic to I_{G/N} and I respectively as H-modules.

As in the preceding proof we have F G/N ≅ F ⊕I. The following proposition is a crucial reduction step. In particular it implies Theorem 2.6 which says that C(G_{n}) and C(ZG ⊗_{ZN} G_{n}) are stably isomorphic as G-fields, so that we can basically study G_{n} at the N-level and induce up.

Proposition 2.4. Let L be a finite ZG-mdoule of exponent p. Suppose we have

(1) An F G-extension of X or X^{−} by L, or of L by X or X^{−}, whose middle term is F G-projective.

(2) A ZG-exact sequence

0 → M → P → L ⊕ Q/mQ → 0

with P and Q stably permutation, where m is a positive integer.

Then the fields C(M) and C(ZG ⊗_{ZN} M) are stably isomorphic as G-fields.

Proof. Tensoring the sequence

0 → M → P → L ⊕ Q/mQ → 0 (s1)

by ZG over ZN we get

0 → ZG ⊗_{ZN} M → ZG ⊗_{ZN} P → F G ⊗_{F N} L ⊕ ZG ⊗_{ZN} Q/mQ → 0. (s2)

We form the sequence

0 → Q → Q → Q/mQ → 0 (s3)

and add it to (s2) to obtain

\begin{aligned}0 & → ZG ⊗_{ZN} M ⊕ Q → ZG ⊗_{ZN} P ⊕ Q \\ & → F G ⊗_{F N} L ⊕ ZG ⊗_{ZN} Q/mQ ⊕ Q/mQ → 0. \end{aligned} (s4)

We have F G⊗_{F N} L ≅ L⊕I ⊗L, and we assume that we have an F G-exact sequence of the form

0 → L → U → X → 0. (1)

The proof extends directly to the other cases. Tensoring (1) by I of F preserves F G-exactness since I is F -free, and we get

0 → L ⊗ I → U ⊗ I → I → 0.

Since U is F G-projective, so is U ⊗I. By looking at the cohomology of the ZG-exact sequence

0 → ZG → ZG → F G → 0,

we see that F G is cohomologically trivial as a ZG-module, therefore so is U ⊗ I. By Proposition 2.3 so is I. Thus L ⊗ I is also ZG-cohomologically trivial. Now consider the exact sequence

(s5) 0 → M′ → M′′ → L ⊗ I → 0 with M′′ free.

Since M′′ and L ⊗ I are cohomologically trivial so is M′, thus M′ is ZG-projective by [BK, Theorem 8.10], and by [EM, Theorem 3.3] it is stably permutation. Now add the sequences (s1), (s5) and

0 → ZG ⊗_{ZN} Q → ZG ⊗_{ZN} Q → ZG ⊗_{ZN} Q/mQ → 0

and form a commutative diagram with (s4),

\xymatrix{ & 0 & 0 \\ 0 → ZG ⊗_{ZN} M ⊕ Q \ar[r] & ZG ⊗_{ZN} P ⊕ Q \ar[r] \ar[u] & \begin{matrix}F G ⊗_{F N} L⊕ \\ ZG ⊗_{ZN} Q/mQ ⊕ Q/mQ → 0 \end{matrix} \ar[u] \\ 0 → ZG ⊗_{ZN} M ⊕ Q \ar[r] \ar[u] & M_{3} \ar[r] \ar[u] & M′′ ⊕ P ⊕ ZG ⊗_{ZN} Q → 0 \ar[u] \\  & M′ ⊕ M ⊕ ZG ⊗_{ZN} Q \ar[r] \ar[u] & M′ ⊕ M ⊕ ZG ⊗_{ZN} Q → 0 \ar[u] \\  & 0 \ar[u] & 0 \ar[u]}

The result now follows by Lemma 2.1, since M′′, P, M′, Q and ZG ⊗_{ZN} Q are stably permutation for this case. A similar argument applies to the other cases.

Corollary 2.5. The fields C(A^{∗}) and C(ZG ⊗_{ZN} A^{∗}) are stably isomorphic as G-fields. So are the fields C(A^{∗−}) and C(ZG ⊗_{ZN} A^{∗−}).

Proof. To prove the first statement of the corollary, we note that there exists an exact sequence

0 → A ⊕ Z → U_{n} → X → 0

where the map from U_{n} to X is given by u_{i} → 1. Tensoring the sequence by A^{∗} over Z we get

0 → B ⊕ A^{∗} → V → T′ → 0,

where T′ = A^{∗}/pA^{∗} as in section 1.

By the remark at the end of section 1 we have an exact sequence

0 → X → U → T′ → 0.

Hence by Proposition 2.4, C(A^{∗}⊕B) and C(ZG⊗_{ZN} (A^{∗}⊕B)) are stably isomorphic as G-fields. The result follows by Lemma 2.1 since B is stably permutation. For the second statement we have the exact sequence

(1) 0 → A^{∗−} ⊕Z^{−} → U_{n}^{−} → T^{′−} → 0, where the map U_{n}^{−} → T^{′−} → 0 is given by U_{n} → A^{∗} → T′ → 0. We also have the sequence 0 → Z ⊕ Z^{−} → ZG/A_{n} → J → 0 where J = Z/2Z.

Tensoring by U_{n} we get

(2) 0 → U_{n}^{−} ⊕ U_{n} → ZG ⊗_{ZA_{n}} U_{n} → J′ → 0 where J′ = U_{n}/2U_{n}, and A_{n} is the alternating group.

Adding U_{n} to the first 2 terms of (1) and combining with (2) we get

0 → A^{∗−} ⊕ Z^{−} ⊕ U_{n} → ZG ⊗_{ZA_{n}} U_{n} → T^{′−} ⊕ J′ → 0

since T^{′−} and J′ are of relatively prime orders.

So C(A^{∗−} ⊕Z^{−} ⊕U_{n}) and C(ZG⊗_{ZN} (A^{∗−} ⊕Z^{−} ⊕U_{n})) are stably isomorphic as G-fields, by Proposition 2.4. Now the result follows by Lemma 2.1 and by observing that Z^{−} is quasi-permutation.

The main theorem is a direct consequence of Proposition 2.4.

Theorem 2.6. The fields C(G_{n}) and C(ZG ⊗_{ZN} G_{n}) are stably isomorphic as G-fields.The center C_{n} is isomorphic to their fixed fields.

Proof. By the remark at the end of section 1, there is an exact sequence

0 → T → U → X → 0.

Applying Proposition 2.4 to the ZG-exact sequence: 0 → G_{n} → B → T → 0 from section 1, we get that C(G_{n}) and C(ZG ⊗_{ZN} G_{n}) are stably isomorphic as Gfields. By [F1, Theorem 3] and [L, Proposition 1.4] we have C_{n} ad C(G_{n})^{G} stably isomorphic.

Theorem 2.7. Let D and R be invertible ZN-lattices. Suppose that R is either 0, or stably permutation and Z_{q}C-projective for all primes q dividing p − 1. If (G_{n} ⊕ R)_{p} ≅ D_{p}, then the fields C(G_{n}) and C(ZG ⊗_{ZN} D) are stably isomorphic as G-fields.

Proof. We have (G_{n} ⊕R)_{p} ≅ D_{p}. By [CR1, Lemma 31.4], this implies the existence of a ZN-exact sequence

0 → G_{n} ⊕ R → D → Y → 0

where Y is a finite ZN-module of order prime to p. Since (G_{n} ⊕ R)_{p} ≅ D_{p}, Q(G_{n} ⊕ R) ≅ QD where Q is the field of rational numbers. It follows directly form the definition of G_{n}, that this lattice is ZC-free. Therefore (G_{n} ⊕ R)_{q} is Z_{q}C-projective for all primes q dividing p − 1, and [B1, Theorem 1.2] implies that so is D_{q}. Since the first 2 terms of the sequence are Z_{q}C-projective for all primes q ≠ p, Y is cohomologically trivial. Therefore we have a ZN-exact sequence

0 → P r → F r → Y → 0

with F r, free. Since Y is cohomologically trivial, so is P r, hence P r is ZNprojective by [BK, Theorem 8.10].

Forming a commutative diagram with the above sequences we obtain

\xymatrix{ & 0 & 0 \\ 0 \ar[r] & G_{n} ⊕ R \ar[r] \ar[u] & D \ar[u] \ar[r] & Y \ar[r] & 0 \\ 0 \ar[r] & G_{n} ⊕ R \ar[r] \ar[u] & M \ar[r] \ar[u] & F r \ar[r] \ar[u] & 0 \\  & P r \ar[r] \ar[u] & P r \ar[u] \\  & 0 \ar[u] & 0 \ar[u]}

This implies that G_{n} ⊕ R ⊕ F r ≅ D ⊕ P r since P r and F r are projective. Inducing up to G we obtain

ZG ⊗_{ZN} (G_{n} ⊕ R ⊕ F r) ≅ ZG ⊗_{ZN} (D ⊕ P r).

Since P r and F r are ZN-projective, the induced modules are ZG-projective and by [EM, Theorem 3.3] they are stably permutation. The result now follows by Lemma 2.1 and Theorem 2.6.

The following lemma will be needed later.

Lemma 2.8. There exist a ZG-lattice W such that G_{n} ⊕ W ≅ V_{n} ⊕ Z(G/H). As ZN-lattices we have G_{n} ⊕ W ≅ ZN^{k} ⊕ Z(N/H) for some positive integer k.

Proof. Since H is cyclic and A ≅ ZH(h−1) as a ZH-module, we have the ZH-exact sequence

0 → Z → ZH → A → 0,

tensoring by ZG over ZH we get

0 → Z(G/H) → ZG → Z(G/H) ⊗ A → 0.

From this sequence we form the following diagram

\xymatrix{ &  & 0 & 0 \\  &  & I ⊗ A \ar[r] \ar[u] & I ⊗ A \ar[u] \\ 0 \ar[r] & Z(G/H) \ar[r] & ZG \ar[r] \ar[u] & Z(G/H) ⊗ A \ar[r] \ar[u] & 0 \\ 0 \ar[r] & Z(G/H) \ar[r] \ar[u] & W \ar[r] \ar[u] & A \ar[r] \ar[u] & 0 \\  &  & 0 \ar[u] & 0 \ar[u]}

where I = I_{G/H}.

For all primes q different from p, dividing the order of G, W_{q} ≅ (A ⊕ Z(G/H))_{q}, since A_{q} is Z_{q}G-invertible by the proof of Proposition 1.1. For the prime p, we have

Z_{p}(G/H) ≅ Z_{p} ⊕ I_{p}.

So

Z_{p}(G/H) ⊗ A_{p} ≅ A_{p} ⊕ I_{p} ⊗ A_{p}.

Since H^{0}(H, ZG/H ⊗ A) ≅ H^{1}(H, ZG/H) = 0, we have

H^{1}(H, W) ≅ H^{0}(H, I ⊗ A) = 0,

and thus W_{p} is Z_{p}H-invertible since H is cyclic by [CT, Corollary 2 and Proposition 2, section 1]. By [B, Theorem 2.1], W is invertible. We have the following commutative diagram

\xymatrix{ &  & 0 & 0 \\ 0 \ar[r] & G_{n} \ar[r] & V_{n} \ar[r] \ar[u] & A \ar[r] \ar[u] & 0 \\ 0 \ar[r] & G_{n} \ar[r] & M \ar[r] \ar[u] & W \ar[r] \ar[u] & 0 \\  &  & ZG/H \ar[r] \ar[u] & ZG/H \ar[u] \\  &  & 0 \ar[u] & 0 \ar[u]}

Since V_{n}, G_{n}, W and ZG/H are invertible, V_{n} ⊕ ZG/H ≅ W ⊕ G_{n} by [CT, Lemma 1, section 1]. The second statement follows from the fact that V_{n} ≅ ZN and Z(G/H) ≅ Z(N/H) ⊕ ZN^{k−1} as ZN-lattices.

3.

Recall that N was defined to be the normalizer of a p-sylow subgroup H of G. So N = N_{G}(H) = H ⋊ C, where C is cyclic of order p − 1, and if we let h and c, generate H and C respectively than c.h = h^{a}, where a is primitive (p − 1)st root of 1 mod p.

Now

Ẑ_{p}N/H = ⨁_{k=1}^{p−1} Z_{k}

where Z_{k} is the Ẑ_{p}N-module of rank 1 with trivial H-action, and such that c acts on 1 as θ^{k}, where θ was chosen to be the primitive (p − 1)-st root of 1 in Ẑ_{p} for which θ ≡ a mod p.

We let X_{k} = Z_{k}/pZ_{k} for k = 1, …, p−1. So X = X_{p−1} is the trivial Z_{p}N-module Z/pZ.

Lemma 3.1. Keeping the definitions of A and A^{∗}. The Z_{p}N-modules A_{p} and A_{p}^{∗} are isomorphic as Z_{p}H-modules. Therefore Z_{p}N ⊗_{Z_{p}H} A ≅ Z_{p}N ⊗_{Z_{p}H} A^{∗}.

Proof. As a ZN-module U_{n} ≅ ZN/C ≅ ZH where u_{i} → h^{i} and c.h = h^{a}. Thus A ≅ ZH(h − 1). Since H is cyclic we have a ZH-exact sequence

0 → Z → ZH → ZH(h − 1) → 0.

Equivalently

0 → Z → ZH → A → 0.

From this sequence and the sequence

0 → Z → ZH → A^{∗} → 0.

We obtain the commutative diagram of ZH-modules.

\xymatrix{ & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Z \ar[r] \ar[d] & ZH \ar[r] \ar[d] & A \ar[r] \ar[d] & 0 \\ 0 \ar[r] & ZH \ar[r] \ar[d] & M \ar[r] \ar[d] & A \ar[r] & 0 \\  & A^{∗} \ar[r] \ar[d] & A^{∗} \ar[d] \\  & 0 & 0}

Thus A^{∗} ⊕ ZH ≅ A ⊕ ZH. By the Krull-Schmidt-Azumaya (K-S-A) Â_{p} ≅ Â_{p}^{∗}, and by [CR1, Theorem 30.17] A_{p} and A_{p}^{∗}.

Notation. For each m dividing p−1, we denote by C_{m} the subgroup of C generated by c^{m}, thus C_{m} has order (p − 1)/m, and index m in C.

We will denote by A_{k} the Ẑ_{p}N-lattice Â_{p}^{∗} ⊗ Z_{k}, for k = 1, …, p − 1.

For each k, we will denote by G_{k} the Ẑ_{p}N-lattice Â_{p}^{∗} ⊗ A_{k}.

Theorem 3.2. We have the following isomorphisms of Ẑ_{p}N-lattices.

(1) A^{∗} ⊗ Z_{1} ≅ Â_{p} as Z_{p}N-lattices.

(2) (Ĝ_{N})_{p} ≅ Â_{p}^{∗} ⊗ A_{2} ≅ G_{2}.

(3) B̂_{p} ≅ G_{1}.

Proof. By Lemma 3.1 we have Z_{p}N ⊗_{Z_{p}H} A ≅ Z_{p}N ⊗_{Z_{p}H} A^{∗}. Thus

⨁_{k=1}^{p−1} A_{k} ≅ ⨁_{k−1}^{p−1} Â_{P} ⊗ Z_{k}.

As an Z_{p}H-module A_{p}^{∗} ≅ Z_{p}[w] where w is a primitive p-th root of 1 on Z_{p}. Therefore Â_{p}^{∗} is indecomposable as a Ẑ_{p}H-module and a fortiori as a Z_{p}N-module. By the K-S-A, this implies that Â_{p} ≅ Â_{p}^{∗} ⊗ Z_{j} for some j. We will show that j = 1. Since Ext_{Ẑ_{p}N}^{1}(Â_{p}, Ẑ_{p}N/H) ≅ Ext_{Ẑ_{p}N}^{1}(Ẑ_{p}N/H, A_{p}^{∗}) ≅ H^{1}(H, A_{p}^{∗}) ≅ Z/p we have Ext_{Ẑ_{p}N}^{1}(Â_{p}, Z_{k}) = 0 for all k except for a unique k for which Ext_{Ẑ_{p}N}^{1}(Â_{p}, Z_{k}) = Z/p. It is easily seen that Z_{k}^{∗} = Hom(Z_{k}, Z_{p}) ≅ Z_{p−1−k}, for k = 1, …, p − 1.

Now taking the cohomology of the sequence 0 → Â_{p} → (Û_{n})_{p} → Ẑ_{p} → 0

we have 0 → Hom_{N}(Ẑ_{p}, Z_{k}) → Hom_{N}((Û_{n})_{p}, Z_{k}) → Hom_{N}(Â_{p}, Z_{k}) → Ext_{N}^{1}(Ẑ_{p}, Z_{k}) = 0 since Z_{k} is invertible. Now if k ≠ p − 1 then Hom_{N}(Ẑ_{p}, Z_{k}) ≅ Z_{k}^{N} = 0, and Hom_{N}((Û_{n})_{p}, Z_{k}) ≅ Z_{k}^{C} ≅ 0, thus Hom_{N}(Â_{p}, Z_{k}) ≅ 0. If k = p − 1, then Hom_{N}(Ẑ_{p}, Z_{k}) ≅ Ẑ_{p}, Hom_{N}((Û_{n})_{p}, Z_{k}) ≅ Ẑ_{p}. Since Hom_{N}(Â_{p}, Z_{k}) is torsion free it must be 0.

Now consider the exact sequence

0 → Z_{k} → Z_{k} → X_{k} → 0.

From this we have 0 → Hom_{N}(Â_{p}, Z_{k}) → Hom_{N}(Â_{p}, Z_{k}) → Hom_{N}(Â_{p}, X_{k}) → Ext_{N}^{1}(Â_{p}, Z_{k}) → 0. Therefore the unique k for which Ext_{N}^{1}(Â_{p}, Z_{k}) ≅ Z/p is the k for which Hom_{N}(A_{p}, X_{k}) = Z/p. Now we have a nonzero Z_{p}N-homomorphism A_{p} → X_{1} given by h − 1 → 1. So k = 1.

Now tensoring the sequence

0 → Ẑ_{p} → (Û_{n})_{p} → A_{p}^{∗} → 0

by Z_{j}, and using the fact that Â_{p} ≅ Â_{p}^{∗} ⊗ Z_{j} we obtain

0 → Z_{j} → (Û_{n})_{p} ⊗ Z_{j} → Â_{p} → 0.

This sequence is not split, hence j = 1. This proves (1).

To prove (2), we tensor both sides of Â_{p} ≅ Â_{p}^{∗} ⊗ Z_{1} by Â_{p} and we get

(Ĝ_{n})_{p} ≅ Â_{p}^{∗} ⊗ Z_{1} ⊗ Â_{p}^{∗} ⊗ Z_{1} ≅ Â_{p}^{∗} ⊗ Â_{p}^{∗} ⊗ Z_{2} ≅ G_{2}.

To prove (3), we tensor Â_{p} ≅ Â_{p}^{∗} ⊗ Z_{1} by A_{p}^{∗} and we get

B_{p} ≅ A_{p}^{∗} ⊗ Â_{p} ≅ Â_{p}^{∗} ⊗ Â_{p}^{∗} ⊗ Z_{1} ≅ G_{1}.

Remark. The lattice A_{2} is an important lattice for the following reason. Consider the exact Z_{p}N-sequence

0 → Ẑ_{p} → Ẑ_{p}H → A_{p}^{∗} → 0.

Tensoring by A_{2} we get

0 → A_{2} → Ẑ_{p}H ⊗ A_{2} → G_{2} → 0.

Since ZH ≅ Z(N/C), Res_{C}^{N} U_{n} ≅ Z ⊕ ZC by Mackey’s subgroup theorem [CR1, Theorem 10.13]. As in the proof of Proposition 1.1, (Û_{n})_{q} ≅ A_{q}^{∗} ⊕ Z_{q} ≅ A_{q} ⊕ Z_{q} for all primes q dividing p−1. By the K-S-A and by [CR1, Theorem 30.17] this implies that A_{q}^{∗} ≅ A_{q} ≅ Z_{q}C. If we let Q be the rationals this implies that QA ≅ QA^{∗} as QC-modules. Since Z_{p}C is a maximal order in QC we have A_{p} ≅ A_{p}^{∗} ≅ Z_{p}C as Z_{p}C-modules by [CR1, Proposition 31.2]. Thus A_{k} ≅ Ẑ_{p}C as Ẑ_{p}C-modules for all k = 1, …, p − 1. Hence Ẑ_{p}H ⊗ A_{2} ≅ Ẑ_{p}N, so ϕ(A_{2}) = [G_{2}]. By the same argument we have ϕ(A_{k}) = [G_{k}] for all k = 1, …, p − 1.

By the above (G_{n})_{q}, (G_{n}^{∗})_{q} and B_{q} are Z_{q}C-free for all primes q dividing p − 1. This fact will be needed later.

We will now use the machinery developed here to give new and very simple proofs for p = 5 and p = 7.

Theorem 3.3. The center of the ring of 5 × 5 generic matrices is stably rational over C.

Proof. Let p = 5; then

Ẑ_{p}N/H = ⨁_{k=1}^{4} Z_{k}

and so

Ẑ_{p}N ⊗_{Ẑ_{p}H} A_{p}^{∗} = A_{p}^{∗} ⊕ A_{1} ⊕ A_{2} ⊕ A_{3}.

Now A_{2} ≅ Â_{p}^{∗} ⊗ Z_{2} ≅ Â_{p}^{∗−}, since Z_{2} ≅ Ẑ_{p}^{−}. Tensoring both sides of A_{2} ≅ Â_{p}^{∗} ⊗ Z_{2} by Â_{p}^{∗} we get by Theorem 3.2

(Ĝ_{N})_{p} ≅ (Ĝ_{N}^{∗−})_{p}.

Now

Ẑ_{p}N ⊗_{Z_{p}HC_{2}} A_{p}^{∗} ≅ Â_{p}^{∗} ⊕ A_{2}.

Tensoring for Â_{p}^{∗} we get

Ẑ_{p}N ⊗_{Z_{p}HC_{2}} (Ĝ_{N}^{∗})_{p} ≅ (Ĝ_{N}^{∗})_{p} ⊕ (Ĝ_{N}^{∗−})_{p}.

Therefore

Z_{p}N ⊗_{Z_{p}HC_{2}} (G_{n}^{∗})_{p} ≅ (G_{n}^{∗})_{p} ⊕ (G_{n})_{p},

and by [CR1, Theorem 30.17]

Z_{p}N ⊗_{Z_{p}HC_{2}} (G_{n}^{∗})_{p} ≅ (G_{n}^{∗})_{p} ⊕ (G_{n})_{p}.

By [CT, Proposition 3, section 1], ϕ((A_{n}^{∗})) = 0 as a ZHC_{2}-module since C_{2} is of order 2. Since G_{n}^{∗} is invertible, this implies that ZN⊗_{ZHC_{2}}G_{n}^{∗} is stably permutation as a ZN-module. Now adding W_{p}^{∗} to both sides of

Z_{p}N ⊗_{Z_{p}HC_{2}} (G_{n}^{∗})_{p} = (G_{n}^{∗})_{p} ⊕ (G_{n})_{p},

we get by Lemma 2.8

Z_{p}N ⊗_{Z_{p}HC_{2}} (G_{n}^{∗})_{p} ⊕ W_{p}^{∗} ≅ (G_{n})_{p} ⊕ Z_{p}N^{k} ⊕ Z_{p}N/H.

Since ZN^{k} ⊕ ZN/H and ZN ⊗_{ZHC_{2}} (G_{n}^{∗}) ⊕ W^{∗} are ZC-free by Lemma 2.8, Theorem 2.7 implies that C(G_{n} ⊕ ZG^{k} ⊕ ZG/H) and C(ZG/N ⊗ W^{∗} ⊕ ZG/HC_{2} ⊗ G_{n}^{∗}) are stably equivalent as G-fields. By Lemma 2.1, C(G_{n}) is stably equivalent to C(ZG/N ⊗ W^{∗}) as G-fields. Now ϕ(A^{∗}) = [G_{n}^{∗}] = Φ(W^{∗}), hence C(ZG/N ⊗ W^{∗}) and C(ZG/N ⊗ A^{∗}) are stably equivalent as G-fields by Lemma 2.1. By Corollary 2.5 these fields are stably equivalent to C(A^{∗}). Therefore by Theorem 2.6, C_{n} is stably isomorphic to C(A^{∗})^{G} which is rational over C, generated by the (n−1)-st elementary symmetric function [BL].

Theorem 3.4. The center of the ring of 7 × 7 generic matrices is stably rational over C.

Proof. The proof is similar to that of p = 5. Here p − 1 = 6.

Ẑ_{p}N/NC_{3} ≅ Ẑ_{p} ⊕ Z_{2} ⊕ Z_{4}

so

Ẑ_{p}N/HC_{3} ⊗ Â_{p}^{∗} ⊕ A_{2} ⊕ A_{4}.

Tensoring by Â_{p}^{∗} we get by Theorem 3.2 and [CR1, Theorem 30.17]

Z_{p}N/HC_{3} ⊗ G_{n}^{∗} ≅ (G_{n}^{∗} ⊕ G_{n} ⊕ B^{−})_{p}

since A_{4} ≅ Â_{p}^{−}. Now consider the exact sequence

0 → B^{−} → ZN/HC_{2} ⊗ B → B → 0.

This sequence splits at all primes different from 2, and for the prime 2, Ext_{N}^{1}(B_{2}, B_{2}^{−}) injects into Ext_{ZC_{3}}^{1}(B_{2}, B_{2}^{−}) = 0 since B_{2} is Z_{2}C-free. So ZN/HC_{2} ⊗B ≅ B ⊕B^{−}, and B^{−} is stably permutation. Also since C_{3} is of order 2, ZN/HC_{3} ⊗ G_{n}^{∗} is stably permutation by [CT, Proposition 3, section 1]. Now adding W_{p}^{∗} to both sides of

Z_{p}N ⊗_{Z_{p}HC_{3}} (G_{n}^{∗})_{p} = (G_{n}^{∗})_{p} ⊕ (G_{n})_{p} ⊕ B_{p}^{−}

we get by Lemma 2.8

Z_{p}N ⊗_{Z_{p}HC_{3}} (G_{n}^{∗})_{p} ⊕ W_{p}^{∗} ≅ (G_{n})_{p} ⊕ Z_{p}N^{k} ⊕ Z_{p}N/H ⊕ B_{p}^{−}.

Since ZN^{k} ⊕ ZN/H ⊕ B^{−} and ZN ⊗_{ZHC_{3}} (G_{n}^{∗}) ⊕ W^{∗} are ZC-free, Theorem 2.7 implies that C(G_{n}⊕ZG^{k} ⊕ZG/H ⊕ZG/H ⊗B^{−}) and C(ZG/N ⊗W^{∗}⊕ZG/HC_{3}⊗ G_{n}^{∗}) are stably equivalent as G-fields. The argument is now the same as for p = 5.

References

[B] E. Beneish, Invertible Modules, J. Algebra 128 (1990), 101–125. MR 91b:20006

[B1] , The Grothendieck ring of invertible modules over nilpotent groups, J. Algebra 159 (1993), 400–418. MR 94i:20009

[BK] K. Brown, Cohomology of groups, Springer-Verlag, New York, 1982. MR 83k:20002

[BL] C. Bessenrodt and L. Le Bruyn, Stable rationality of certain P GL_{n}-quotients, Invent. Math. 104 (1991), 179–199. MR 92m:14060

[CT] J.-L. Colliot-Thelene et J.-J. Sansuc, La R-equivalence sur les tores, Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), 175–230. MR 56:8576

[CR1] C. W. Curtis and I. Reiner, Methods of Representation Theory, vol. 1, Wiley, New York, 1981. MR 90k:20001

[CR2] , Methods of Representation Theory, vol. 2, Wiley, New York, 1987. MR 88f:20002 [EM] S. Endo and T. Miyata, On the projective class group of finite groups, Osaka J. Math. 13

(1976), 109–122. MR 53:13315

[F1] E. Formanek, The center of the ring of 3 × 3 generic matrices, Linear and Multilinear Algebra 7 (1979), 203–212. MR 80h:16019

[F2] , The center of the ring of 4 × 4 generic matrices, J. Algebra 62 (1980), 304–319. MR 81g:15032

[L] H. W. Lenstra, Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299–325. MR 50:289

[LL] L. Le Bruyn, Centers of generic division algebras, the rationality problem 1965–1990, Israel Journal of Math. 76 (1991), 97–111. MR 93f:16024

[P] C. Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei 8 (1967), 237–255. MR 37:256

[SD] D. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), 165–215. MR 85j:13008

[SR] R. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148–158. MR 39:5532

[S] J. Sylvester, On the involution of two matrices of second order, Southport: British Assoc. Report (1883), 430–432.

Department of Mathematics, University of Wisconsin-Parkside, Kenosha, Wisconsin 53141-2000